--- trunk/tengDissertation/LiquidCrystal.tex 2006/06/22 05:23:06 2877 +++ trunk/tengDissertation/LiquidCrystal.tex 2006/07/17 20:01:05 2941 @@ -2,55 +2,52 @@ Long range orientational order is one of the most fund \section{\label{liquidCrystalSection:introduction}Introduction} -Long range orientational order is one of the most fundamental -properties of liquid crystal mesophases. This orientational -anisotropy of the macroscopic phases originates in the shape -anisotropy of the constituent molecules. Among these anisotropy -mesogens, rod-like (calamitic) and disk-like molecules have been -exploited in great detail in the last two decades\cite{Huh2004}. -Typically, these mesogens consist of a rigid aromatic core and one -or more attached aliphatic chains. For short chain molecules, only -nematic phases, in which positional order is limited or absent, can -be observed, because the entropy of mixing different parts of the -mesogens is paramount to the dispersion interaction. In contrast, -formation of the one dimension lamellar sematic phase in rod-like -molecules with sufficiently long aliphatic chains has been reported, -as well as the segregation phenomena in disk-like molecules. +Rod-like (calamitic) and disk-like anisotropy liquid crystals have +been investigated in great detail in the last two +decades.\cite{Huh2004} Typically, these mesogens consist of a rigid +aromatic core and one or more attached aliphatic chains. For short +chain molecules, only nematic phases, in which positional order is +limited or absent, can be observed, because the entropy of mixing +different parts of the mesogens is larger than the dispersion +interaction. In contrast, formation of one dimension lamellar +smectic phase in rod-like molecules with sufficiently long aliphatic +chains has been reported, as well as the segregation phenomena in +disk-like molecules.\cite{McMillan1971} Recently, banana-shaped or +bent-core liquid crystals have became one of the most active +research areas in mesogenic materials and supramolecular +chemistry.\cite{Niori1996, Link1997, Pelzl1999} Unlike rods and +disks, the polarity and biaxiality of the banana-shaped molecules +allow the molecules organize into a variety of novel liquid +crystalline phases which show interesting material properties. Of +particular interest is the spontaneous formation of macroscopic +chiral layers from achiral banana-shaped molecules, where polar +molecule orientational ordering exhibited layered plane as well as +the tilted arrangement of the molecules relative to the polar axis. +As a consequence of supramolecular chirality, the spontaneous +polarization arises in ferroelectric (FE) and antiferroelectic (AF) +switching of smectic liquid crystal phases, demonstrating some +promising applications in second-order nonlinear optical devices. +The most widely investigated mesophase formed by banana-shaped +moleculed is the $\text{B}_2$ phase, which is also referred to as +$\text{SmCP}$.\cite{Link1997} Of the most important discoveries in +this tilt lamellar phase is the four distinct packing arrangements +(two conglomerates and two macroscopic racemates), which depend on +the tilt direction and the polar direction of the molecule in +adjacent layer (see Fig.~\ref{LCFig:SMCP}).\cite{Link1997} -Recently, the banana-shaped or bent-core liquid crystal have became -one of the most active research areas in mesogenic materials and -supramolecular chemistry\cite{Niori1996, Link1997, Pelzl1999}. -Unlike rods and disks, the polarity and biaxiality of the -banana-shaped molecules allow the molecules organize into a variety -of novel liquid crystalline phases which show interesting material -properties. Of particular interest is the spontaneous formation of -macroscopic chiral layers from achiral banana-shaped molecules, -where polar molecule orientational ordering is shown within the -layer plane as well as the tilted arrangement of the molecules -relative to the polar axis. As a consequence of supramolecular -chirality, the spontaneous polarization arises in ferroelectric (FE) -and antiferroelectic (AF) switching of smectic liquid crystal -phases, demonstrating some promising applications in second-order -nonlinear optical devices. The most widely investigated mesophase -formed by banana-shaped moleculed is the $\text{B}_2$ phase, which -is also referred to as $\text{SmCP}$\cite{Link1997}. Of the most -important discover in this tilt lamellar phase is the four distinct -packing arrangements (two conglomerates and two macroscopic -racemates), which depend on the tilt direction and the polar -direction of the molecule in adjacent layer (see -Fig.~\ref{LCFig:SMCP}). - \begin{figure} \centering \includegraphics[width=\linewidth]{smcp.eps} -\caption[] -{} +\caption[SmCP Phase Packing] {Four possible SmCP phase packings that +are characterized by the relative tilt direction(A and S refer an +anticlinic tilt or a synclinic ) and the polarization orientation (A +and F represent antiferroelectric or ferroelectric polar order).} \label{LCFig:SMCP} \end{figure} Many liquid crystal synthesis experiments suggest that the occurrence of polarity and chirality strongly relies on the -molecular structure and intermolecular interaction\cite{Reddy2006}. +molecular structure and intermolecular interaction.\cite{Reddy2006} From a theoretical point of view, it is of fundamental interest to study the structural properties of liquid crystal phases formed by banana-shaped molecules and understand their connection to the @@ -60,49 +57,47 @@ smectic arrangements\cite{Cook2000, Lansac2001}, as we ordering and phase behavior, and hence improve the development of new experiments and theories. In the last two decades, all-atom models have been adopted to investigate the structural properties of -smectic arrangements\cite{Cook2000, Lansac2001}, as well as other +smectic arrangements,\cite{Cook2000, Lansac2001} as well as other bulk properties, such as rotational viscosity and flexoelectric -coefficients\cite{Cheung2002, Cheung2004}. However, due to the -limitation of time scale required for phase transition and the +coefficients.\cite{Cheung2002, Cheung2004} However, due to the +limitation of time scales required for phase transition and the length scale required for representing bulk behavior, -models\cite{Perram1985, Gay1981}, which are based on the observation +models,\cite{Perram1985, Gay1981} which are based on the observation that liquid crystal order is exhibited by a range of non-molecular -bodies with high shape anisotropies, became the dominant models in -the field of liquid crystal phase behavior. Previous simulation -studies using hard spherocylinder dimer model\cite{Camp1999} produce -nematic phases, while hard rod simulation studies identified a -Landau point\cite{Bates2005}, at which the isotropic phase undergoes -a direct transition to the biaxial nematic, as well as some possible -liquid crystal phases\cite{Lansac2003}. Other anisotropic models -using Gay-Berne(GB) potential, which produce interactions that favor -local alignment, give the evidence of the novel packing arrangements -of bent-core molecules\cite{Memmer2002,Orlandi2006}. +bodies with high shape anisotropies, have become the dominant models +in the field of liquid crystal phase behavior. Previous simulation +studies using a hard spherocylinder dimer model\cite{Camp1999} +produced nematic phases, while hard rod simulation studies +identified a direct transition to the biaxial nematic and other +possible liquid crystal phases.\cite{Lansac2003} Other anisotropic +models using the Gay-Berne(GB) potential, which produces +interactions that favor local alignment, give evidence of the novel +packing arrangements of bent-core molecules.\cite{Memmer2002} Experimental studies by Levelut {\it et al.}~\cite{Levelut1981} revealed that terminal cyano or nitro groups usually induce permanent longitudinal dipole moments, which affect the phase -behavior considerably. A series of theoretical studies also drawn -equivalent conclusions. Monte Carlo studies of the GB potential with -fixed longitudinal dipoles (i.e. pointed along the principal axis of -rotation) were shown to enhance smectic phase -stability~\cite{Berardi1996,Satoh1996}. Molecular simulation of GB +behavior considerably. Equivalent conclusions have also been drawn +from a series of theoretical studies. Monte Carlo studies of the GB +potential with fixed longitudinal dipoles (i.e. pointed along the +principal axis of rotation) were shown to enhance smectic phase +stability.\cite{Berardi1996,Satoh1996} Molecular simulation of GB ellipsoids with transverse dipoles at the terminus of the molecule also demonstrated that partial striped bilayer structures were -developed from the smectic phase ~\cite{Berardi1996}. More +developed from the smectic phase.~\cite{Berardi1996} More significant effects have been shown by including multiple electrostatic moments. Adding longitudinal point quadrupole moments to rod-shaped GB mesogens, Withers \textit{et al} induced tilted -smectic behaviour in the molecular system~\cite{Withers2003}. Thus, +smectic behaviour in the molecular system.~\cite{Withers2003} Thus, it is clear that many liquid-crystal forming molecules, specially, bent-core molecules, could be modeled more accurately by incorporating electrostatic interaction. -In this chapter, we consider system consisting of banana-shaped -molecule represented by three rigid GB particles with one or two -point dipoles at different location. Performing a series of -molecular dynamics simulations, we explore the structural properties -of tilted smectic phases as well as the effect of electrostatic -interactions. +In this chapter, we consider a system consisting of banana-shaped +molecule represented by three rigid GB particles with two point +dipoles. Performing a series of molecular dynamics simulations, we +explore the structural properties of tilted smectic phases as well +as the effect of electrostatic interactions. \section{\label{liquidCrystalSection:model}Model} @@ -129,11 +124,11 @@ orientation of two molecules $i$ and $j$ separated by } \right] \label{LCEquation:gb} \end{equation} where $\hat u_i,\hat u_j$ are unit vectors specifying the -orientation of two molecules $i$ and $j$ separated by intermolecular -vector $r_{ij}$. $\hat r_{ij}$ is the unit vector along the -intermolecular vector. A schematic diagram of the orientation -vectors is shown in Fig.\ref{LCFigure:GBScheme}. The functional form -for $\sigma$ is given by +orientation of two ellipsoids $i$ and $j$ separated by +intermolecular vector $r_{ij}$. $\hat r_{ij}$ is the unit vector +along the inter-ellipsoid vector. A schematic diagram of the +orientation vectors is shown in Fig.\ref{LCFigure:GBScheme}. The +functional form for $\sigma$ is given by \begin{equation} \sigma (\hat u_i ,\hat u_i ,\hat r_{ij} ) = \sigma _0 \left[ {1 - \frac{\chi }{2}\left( {\frac{{(\hat r_{ij} \cdot \hat u_i + \hat @@ -181,23 +176,18 @@ ratio between \textit{end-to-end} well depth $\epsilon \begin{figure} \centering \includegraphics[width=\linewidth]{banana.eps} -\caption[]{} \label{LCFig:BananaMolecule} +\caption[Schematic representation of a typical banana shaped +molecule]{Schematic representation of a typical banana shaped +molecule.} \label{LCFig:BananaMolecule} \end{figure} - -%\begin{figure} -%\centering -%\includegraphics[width=\linewidth]{bananGB.eps} -%\caption[]{} \label{LCFigure:BananaGB} -%\end{figure} - \begin{figure} \centering \includegraphics[width=\linewidth]{gb_scheme.eps} -\caption[]{Schematic diagram showing definitions of the orientation -vectors for a pair of Gay-Berne molecules} -\label{LCFigure:GBScheme} +\caption[Schematic diagram showing definitions of the orientation +vectors for a pair of Gay-Berne molecules]{Schematic diagram showing +definitions of the orientation vectors for a pair of Gay-Berne +ellipsoids} \label{LCFigure:GBScheme} \end{figure} - To account for the permanent dipolar interactions, there should be an electrostatic interaction term of the form \begin{equation} @@ -208,19 +198,20 @@ where $\epsilon _{fs}$ is the permittivity of free spa \end{equation} where $\epsilon _{fs}$ is the permittivity of free space. -\section{Computational Methodology} +\section{Results and Discussion} -A series of molecular dynamics simulations were perform to study the +A series of molecular dynamics simulations were performed to study the phase behavior of banana shaped liquid crystals. In each simulation, -every banana shaped molecule has been represented three GB particles -which is characterized by $\mu = 1,~ \nu = 2, +every banana shaped molecule has been represented by three GB +particles which is characterized by $\mu = 1,~ \nu = 2, ~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. All of the simulations begin with same equilibrated isotropic configuration where 1024 molecules without dipoles were confined in a $160\times 160 \times 120$ box. After the dipolar interactions are switched on, 2~ns NPTi cooling run with themostat of 2~ps and barostat of 50~ps were used to equilibrate the system to desired -temperature and pressure. +temperature and pressure. NPTi Production runs last for 40~ns with +time step of 20~fs. \subsection{Order Parameters} @@ -228,8 +219,8 @@ the largest eigen value obtained by diagonalizing the calculated various order parameters and correlation functions. Particulary, the $P_2$ order parameter allows us to estimate average alignment along the director axis $Z$ which can be identified from -the largest eigen value obtained by diagonalizing the order -parameter tensor +the largest eigenvalue obtained by diagonalizing the order parameter +tensor \begin{equation} \overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % \begin{pmatrix} % @@ -237,7 +228,7 @@ parameter tensor u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % \end{pmatrix}, -\label{lipidEq:po1} +\label{lipidEq:p2} \end{equation} where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector $\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole @@ -247,53 +238,125 @@ In addition to the $P_2$ order parameter, $ R_{2,2}^2$ \langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. \label{lipidEq:po3} \end{equation} -In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order -parameter for biaxial phase is introduced to describe the ordering -in the plane orthogonal to the director by -\begin{equation} -R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot -Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle -\end{equation} -where $X$, $Y$ and $Z$ are axis of the director frame. +%In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order +%parameter for biaxial phase is introduced to describe the ordering +%in the plane orthogonal to the director by +%\begin{equation} +%R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot +%Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle +%\end{equation} +%where $X$, $Y$ and $Z$ are axis of the director frame. +The unit vector for the banana shaped molecule was defined by the +principle aixs of its middle GB particle. The $P_2$ order parameters +for the bent-core liquid crystal at different temperature are +summarized in Table~\ref{liquidCrystal:p2} which identifies a phase +transition temperature range. -\subsection{Structure Properties} +\begin{table} +\caption{LIQUID CRYSTAL STRUCTURAL PROPERTIES AS A FUNCTION OF +TEMPERATURE} \label{liquidCrystal:p2} +\begin{center} +\begin{tabular}{cccccc} +\hline +Temperature (K) & 420 & 440 & 460 & 480 & 600\\ +\hline +$\langle P_2\rangle$ & 0.984 & 0.982 & 0.975 & 0.967 & 0.067\\ +\hline +\end{tabular} +\end{center} +\end{table} -It is more important to show the density correlation along the -director +\subsection{Structural Properties} + +The molecular organization obtained at temperature $T = 460K$ (below +transition temperature) is shown in Figure~\ref{LCFigure:snapshot}. +The diagonal view in Fig~\ref{LCFigure:snapshot}(a) shows the +stacking of the banana shaped molecules while the side view in +Figure~\ref{LCFigure:snapshot}(b) demonstrates formation of a +chevron structure. The first peak of the radial distribution +function $g(r)$ in Fig.~\ref{LCFigure:gofrz}(a) shows that the +minimum distance for two in plane banana shaped molecules is 4.9 +\AA, while the second split peak implies the biaxial packing. It is +also important to show the density correlation along the director +which is given by : \begin{equation} -g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho -\end{equation}, -where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame -and $R$ is the radius of the cylindrical sampling region. +g(z) =\frac{1}{\pi R^{2} \rho}< \delta (z-z_{ij})>_{ij}, +\end{equation} +where $ z_{ij} = r_{ij} \cdot \hat Z $ was measured in the +director frame and $R$ is the radius of the cylindrical sampling +region. The oscillation in density plot along the director in +Fig.~\ref{LCFigure:gofrz}(b) implies the existence of the layered +structure, and the peak at 27 $\rm{\AA}$ is attributed to a defect in the +system. \subsection{Rotational Invariants} As a useful set of correlation functions to describe position-orientation correlation, rotation invariants were first applied in a spherical symmetric system to study x-ray and light -scatting\cite{Blum1971}. Latterly, expansion of the orientation pair +scatting.\cite{Blum1972} Latterly, expansion of the orientation pair correlation in terms of rotation invariant for molecules of -arbitrary shape was introduce by Stone\cite{Stone1978} and adopted -by other researchers in liquid crystal studies\cite{Berardi2000}. +arbitrary shape has been introduced by Stone\cite{Stone1978} and +adopted by other researchers in liquid crystal +studies.\cite{Berardi2003} In order to study the correlation between +biaxiality and molecular separation distance $r$, we calculate a +rotational invariant function $S_{22}^{220} (r)$, which is given by +: +\begin{eqnarray} +S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }} \left< \delta (r - +r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j +)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j +)^2 ) \right. \notag \\ + & & \left. - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - +2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j )) \right>. +\end{eqnarray} +The long range behavior of second rank orientational correlation +$S_{22}^{220} (r)$ in Fig~\ref{LCFigure:S22220} also confirm the +biaxiality of the system. -\begin{equation} -S_{22}^{220} (r) & = & \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r -- r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat -y_j )^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat -y_j )^2 ) - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) -- 2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} -\right\rangle -\end{equation} +\begin{figure} +\centering +\includegraphics[width=4.5in]{snapshot.eps} +\caption[Snapshot of the molecular organization in the layered phase +formed at temperature T = 460K and pressure P = 1 atm]{Snapshot of +the molecular organization in the layered phase formed at +temperature T = 460K and pressure P = 1 atm. (a) diagonal view; (b) +side view.} \label{LCFigure:snapshot} +\end{figure} -\begin{equation} -S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle -{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot -\hat z_j \times \hat r_{ij} ))} \right\rangle -\end{equation} +\begin{figure} +\centering +\includegraphics[width=\linewidth]{gofr_gofz.eps} +\caption[Correlation Functions of a Bent-core Liquid Crystal System +at Temperature T = 460K and Pressure P = 10 atm]{Correlation +Functions of a Bent-core Liquid Crystal System at Temperature T = +460K and Pressure P = 10 atm. (a) radial correlation function +$g(r)$; and (b) density along the director $g(z)$.} +\label{LCFigure:gofrz} +\end{figure} -\section{Results and Conclusion} -\label{sec:results and conclusion} +\begin{figure} +\centering +\includegraphics[width=\linewidth]{s22_220.eps} +\caption[Average orientational correlation Correlation Functions of +a Bent-core Liquid Crystal System at Temperature T = 460K and +Pressure P = 10 atm]{Average orientational correlation Correlation +Functions of a Bent-core Liquid Crystal System at Temperature T = +460K and Pressure P = 10 atm.} \label{LCFigure:S22220} +\end{figure} -To investigate the molecular organization behavior due to different -dipolar orientation and position with respect to the center of the -molecule, +\section{Conclusion} + +We have presented a simple dipolar three-site GB model for banana +shaped molecules which are capable of forming smectic phases from +isotropic configuration. Various order parameters and correlation +functions were used to characterized the structural properties of +these smectic phase. However, the forming layered structure still +had some defects because of the mismatching between the layer +structure spacing and the shape of simulation box. This mismatching +can be broken by using NPTf integrator in further simulations. The +role of terminal chain in controlling transition temperatures and +the type of mesophase formed have been studied +extensively.\cite{Pelzl1999} The lack of flexibility in our model +due to the missing terminal chains could explain the fact that we +did not find evidence of chirality.